Most mathematicians consider themselves, at least to some degree, to
be both educators and researchers. Yet few embrace or even respect the
subject at the intersection of these fields: research in mathematics
education. Mathematicians rarely apply their own logical acumen to
assessment of teaching and learning, nor do they often read the vast and
growing literature in educational research. They are conflicted about
evaluating the professional work of researchers in mathematics education
and often resist pressure to publish (or even review) educational
research in mainstream professional journals. Few look to educational
research for insight when seeking to improve their own teaching.

Why is this? Many researchers in mathematics education believe their
field is the victim of prejudice by a conservative old guard who reject
anything that does not fit the paradigm of classical scientific
research. Mathematics departments have a long history of resisting new
interdisciplinary fields (e.g., statistics, operations research, applied
mathematics, computer science). Indeed, according to a recent NSF
review committee, mathematicians still disdain connections with outside
fields, even when such isolation hurts their self-interest. Research in
mathematics education may well be just the latest victim of
mathematicians' hubris.

On the other hand, perhaps the phrase "research in mathematics
education" is simply a pretension, if not an oxymoron. Is mathematics
education even susceptible to research? What might be the aims of such
research? What are the objects of study? What are the chief questions?
What are the main theories? What are the key results? What are the
criteria? What are the important applications? Critics of research in
mathematics education point to the paucity of answers to these questions
as evidence that what we are dealing with here is more like a political
movement than a scientific discipline.

In response to these and related concerns, the International
Commission on Mathematical Instruction (ICMI) convened experts in
mathematics education research from around the world and asked them to
clarify the identity of their discipline. The purpose of this study
conference, held in May 1994 at the University of Maryland, was not to
describe the state of the art but to identify perspectives, goals,
problems, and research methodologies. The two volumes under review
contain reports from this study conference together with several dozen
papers expanded from conference presentations.

So what do these international experts tell us about their own field?
Pretty much the same thing as the skeptical mathematicians. Indeed,
the most striking conclusion of this ICMI study is that in spite of much
thoughtful work by individual researchers, there is no agreement among
leaders in the field about goals of research, important questions,
objects of study, methods of investigation, criteria for evaluation,
significant results, major theories, or usefulness of results. The
papers in these two volumesfive working group reports, 33 expert
papers, and the editors' summary (significantly titled "Continuing the
Search")document a field in disarray, a field whose high hopes for
a science of education have been overwhelmed by complexity and drowned
in a sea of competing theories.

The title of the volume virtually proclaims a crisis in identity.
What other academic field would devote such effort to investigating
whether or not it is a research discipline? The standards are set
forthrightly by one of the study's working groups: Research is "a craft
practiced by scholarly groups whose members have agreed in a broad sense
on what procedures are to be followed and on the criteria for acceptable
work" (Working Group 2, p. 16). Judging from the papers in this volume,
research in mathematics education falls far short of meeting these two
standards.

Despite arguments about the relative importance of different
subdisciplines, mathematicians for the most part agree on procedures of
research and criteria of validity for mathematical results. For
theorems, logical proofs are required; for applications and algorithms,
demonstrated utility suffices until rigorous proofs can be found; and
for theories, broad explanatory power is expected. Scientists operate
in a domain in which theoretical models must be tested against worldly
events. Yet all scientists agree on the requirements of reproducible
procedures and empirical verification.

In contrast, researchers in mathematics education not only disagree
on criteria for acceptable work, they even disagree on the need for
criteria. Some refuse to adopt any criteria. They propose instead "to
define specific approaches and methods for each problem separately" (A.
Sierpinska & J. Kilpatrick, p. 540). Some urge adoption of
traditional criteria (K. Hart, p. 411) whereas others support a
"practical perspective" that recognizes the primacy of idiosyncratic
personal views of editors and reviewers (G. Hanna, p. 401). "The study
has shown," report the despairing editors (p. 540), "that mathematics
educators generally feel uncomfortable with the idea of establishing ...
a set of criteria for assessing the quality of mathematics education
research."

OK. Perhaps it is too much to expect widespread agreement on the
criteria for research in a field as complex and amorphous as mathematics
education. But mathematics educators cannot even agree on the nature of
mathematics. Although mathematics is at the heart of mathematics
education, it turns out that educators' mathematics is not
mathematicians' mathematics. Indeed, the question of "what is
mathematics" is debated throughout these two volumes almost as if
Courant and Robbins (and their many successors) had never answered the
question. The consequences for education are profound. "If it is
unclear as to what is mathematics, what are its main achievements, and
what constitutes performance in it, then what hope is there of teaching
it in a clear way or more effectively?" (R. Brown, p. 459).

To a mathematician, mathematics is singulara Platonic paradigm
in which there are simple, unquestionable criteria for distinguishing
right from wrong and true from false. But to mathematics educators,
mathematics is plural. Mathematics, among other things, offers a lens
through which one can look at the world. In mathematics education the
direction is reversedone looks at mathematics through the lens of
learners (and teachers). Thus "a multitude of views on mathematics will
result," even a multitude of "personal mathematics" (W. Dörfler, p.
13). In mathematics education, post-modernism has replaced Platonism,
contextual interpretation has replaced absolute truth, and utility has
replaced correctness as the standard of value (A. Sfard, p. 491). No
wonder educators repeatedly cite non-Platonists such as Imre Lakatos,
Reuben Hersch, and Thomas Tymoczko as authorities on the nature of
mathematics rather than, say, Keith Devlin, Morris Kline, or Ian
Stewart, each of whom has written extensively on this topic.

This difference in the conception of mathematics has important
consequences for mathematics education. Mathematicians view their
mathematics as the real thing, in contrast to other people's versions
that mathematicians believe are but imperfect imitations.
Nonmathematicians, especially students, harbor nonstandard ideas
("misconceptions") that, "like road accidents," could be avoided through
better teaching. In mathematicians' Platonic world view, "idiosyncratic
student conceptions" are the result of "failed attempts to convey ...
what true mathematics is all about" (A. Sfard, p. 496).

In contrast, mathematics educators' view of mathematics is mediated
by the oftentimes strained relation between mathematics and its
learnerswhat one author called the "pragmatics" of mathematics.
This relation of mathematics to learner "requires us to broaden the
notion of mathematics to encompass mathematical practices that are
relevant for society at large but [are practices] other than those of
university mathematics" (Working Group 1, p. 11). The inherent
incompatibility between this pluralistic post-modern view and the naive
Platonism of working mathematicians yields systems of beliefs "as
distinct as those which separate incommensurable scientific paradigms
... or different religions" (A. Sfard, p. 505). "Judging one from
within the other is absurd."

The endless cycle of fashion in mathematics educationfirst new
math, then back to basics, then problem solving, now
constructivismleads many observers to question whether the subject
harbors any enduring principles or convictions. The debate in these
volumes does little to mitigate this concern. Dramatically opposing
views on almost every topic are more the norm than the exception.

For example, some argue that all mathematical knowledge comes into
being "in contexts that are real, concrete, or experimental" (J.
Confrey, p. 12), whereas others hold that important parts of mathematics
must be learned "not in authentic but in artificial contexts" (G.
Vergnaud, p. 12). Although many believe in the scientific basis for
research in mathematics education, others argue that such knowledge is
just "a combination of statements of belief and their justification
which relate to the [environment] in which the student is operating" (S.
Lerman & R. Lins, p. 342). In fact, some are so concerned about the
disharmony of the English term "mathematics education" that they prefer
using didactique des matématiques.

A chief worry of many authors is whether their field is worthy. Is
it a true scientific discipline (whose products are theories of
learning) or a design science (whose products are curricula, texts, and
software)? Believing in the former as the path to both knowledge and
stature, many U.S. mathematics educators undertook comparative
statistical research that they expected would lead to "grand theories
that would enable mathematics education to become a science" (N.
Ellerton & M. Clements, p. 157). This strategy, advocated by Ed
Begle in 1966 at the First International Congress on Mathematics
Education, was based on a view of mathematics education as an
experimental science similar to medicine or agriculture. Unfortunately,
not long after the challenge was laid down, "doubts were being expressed
about the capability of education to become a science at all" (R. Mura,
p. 114).

This American infatuation with statistical research has now been
moderated by an infusion of observational and inferential methodologies
drawn more from anthropology than from agriculture. Children, it turns
out, are more like people than like plants. Problems in mathematics
education are inherently too complex for simple statistical techniques,
requiring instead methodologies from many disciplines including
epistemology, psychology, sociology, philosophy, neuroscience, and
mathematics itself.

Mathematicians tend to believe that only when mathematics education
research produces a "theory of mathematical thinking that convincingly
explains observed phenomena" (S. Amitsur, p. 455) will it become a true
academic discipline on a par with other sciences that produce robust
theories with broad explanatory power (e.g., evolution in biology,
thermodynamics in chemistry, relativity in physics). But are such
theories likely to emerge in a field that defiantly refuses to accept
even its own past accomplishments? "There is no way to resolve
[questions about research in mathematics education] once and for all in
the way, for example, one might prove a theorem. Each generation must
address anew what doing research in mathematics education is all about"
(A. Sierpinska & J. Kilpatrick, p. 527). Such sentiments more aptly
describe pop culture than a "research domain" of the title quest.

One reason research experts agree on so little is that education is
socially and culturally situated. Thus, for instance, the energy that
U.S. researchers devote to problem solving is of little interest to
Japanese educators because problem solving is already a routine part of
their educational practice (Y. Sekiguchi, p. 392), whereas the
individualistic focus of the Piagetian psychological literature that has
so dominated mathematics education in the West has little in common with
Vygotsky's "radically distinct" social constructivism that has
influenced Eastern Europe (S. Lerman, p. 347). The international
community of mathematics education research is not like that of
mathematics or science. In the latter, "research is internationally
shared and international interests are overriding," whereas in the
former "each country has its domestic educational problems and
interests, and they shape its research practice significantly" (Y.
Sekiguchi, p. 391).

Several study participants noted that most references in the research
literature in mathematics education point to the research school to
which the author belongs. This intellectual myopia, they argue, is due
not to language barriers or lack of good will but to "true
incompatibilities" between different research traditions. "Results
obtained by different research schools are very difficult to compare,
and researchers prefer to stay within a strictly homogeneous reference
system" (P. Boero & J. Szendrei, p. 207). The result is intense
intellectual parochialism in which arguments over epistemology or
methodology that rage among researchers in one country may be looked on
with "amusement, disdain, or incomprehension" by researchers elsewhere
(A. Sierpinska & J. Kilpatrick, p. 546).

To study. "Research in mathematics education is the
intentionally controlled examination of issues within and related to the
learning and teaching of mathematics through a process of inquiry that
leads to the production of (provisional) knowledge both about the
objects of the inquiry and the means of carrying out that inquiry" (G.
Hatch & C. Shiu, p. 297).

To understand. "Developing understanding satisfies both
fundamental (theoretic) and practical aims simultaneously. Research
efforts that do not aim to understand satisfy neither of these goals and
are not worth making" (J. Hiebert, p. 141).

To create knowledge. "The goal for research in mathematics
education should be to produce new knowledge about the teaching and
learning of mathematics" (T. Romberg, p. 379).

To improve education. "Among the more pragmatic aims would
be the improvement of teaching practice as well as of student
understanding and performance." (Discussion Document, p. 5).

Opinions about the desired results of research in mathematics
education are equally diverse, if not sometimes incompatible.
Mathematicians expect verifiable studies that offer the possibility of
improving teaching and learning. Some educators argue that change in
peopletransformation in the "being" of the researchers
themselvesis the most important product. "It is their questions
that change, their sensitivities that develop, ... their perspectives
that alter. In short, it is their being that develops" (J. Mason, p.
357). Others argue for empirically tested teaching units based on
"fundamental theoretical principles. ... Such units are the most
efficient carriers of innovation and are well suited to bridge the gap
between theory and practice" (E. Wittmann, p. 99).

Of course mathematicians, if asked about the desired results of their
own field, would also give a number of different answers (e.g., proofs,
theorems, applications, algorithms, concepts, theories). However, most
mathematicians would embrace all these options as significant and
worthwhile results. In contrast, the papers in this ICMI study give the
distinct impression of a field Balkanized by conflicting ideologies in
which each researcher argues for idiosyncratic definitions and goals.
The result is intellectual nihilism rooted in the "continuing and
persistent doubt" of not knowing. "Instead of searching desperately for
secure ground on which to stand firm, it is possible to accept tension
between knowing and not knowing as a productive and inescapable source
of energy and security" (J. Mason, p. 359).

Behind the title of this study is a faint hint of worry that the
emperor of educational research may have no clothes. Physicists study
energy and matter; chemists, molecules; biologists, life;
mathematicians, patterns. But what do researchers in mathematics
education study? According to the very comprehensive index to these
volumes, they study contexts, cultures, and curricula; discourse,
language, and meaning; teaching, learning, and curricula; models,
paradigms, and philosophies; situations, practices, and theories;
knowledge, understanding, and meaning; radical, social, scientific, and
trivial constructivism; as well as such esoterica as acculturation,
didactic phenomenology, educational ecologies, epistemological
empowerment, hermeneutics, intersubjectivity, metonymy, ontology,
semantic fields, semiotic models, and situated cognition. (Mathematics,
it seems, does not have a corner on the market of abstruse terminology.)
Unfortunately, nothing in these volumes suggests that this profusion of
vocabulary is sufficient to clothe the emperor.

From this thicket of meaning-challenged words and related
disputations about goals, aims, methods, criteria, and results emerges a
single irony that seems to enjoy widespread assent: Research has had
essentially no impact on the practice of mathematics education. On this
broad indictment, researchers and their critics agreealthough, as
usual, they do not agree on the cause. Some blame incompatibilities
between researchers' aims and the reality of mathematics education (P.
Boero & J. Szendrei, p. 198) or the widespread tendency to
generalize results beyond their field of validity (Working Group 4, p.
27). Others cite the inclination of researchers to talk only to each
other. "The lack of relationship between research and practice is well
documented" (A. Bishop, p. 35).

Almost everyone seems to recognize a significant divergence of
interest between researchers and teachers. "Mathematics teachers tend
to expect ... specific prescriptions to solve problems they experience
in their classrooms, whereas researchers tend to address problems raised
in research communities that are not always directly related to
teachers' concerns" (Y. Sekiguchi, p. 391). "Failure is the only
possible outcome for any approach in which researchers hand their
results to curriculum developers and teachers who are then expected to
apply them in their practices" (J. Mason, p. 375).

Even those outside the relatively inbred community of mathematics
education researchers have picked up the scent of irrelevance.
Mathematicians, struggling to cope with the "democratization" of
teaching, "do not find in the results of didactic research the means to
remedy the problems that, in their opinion, this research should
provide" (M. Artigue, p. 484). University presidents talk about closing
their schools of education; colleagues find most researchers' papers
uninteresting ("they deal with marginal details," S. Amitsur, p. 448);
and teachers cannot synthesize scattered results into useful forms (A.
Bishop, p. 42). Even the public now sees the activities of the
mathematics education research community to be "at best marginal and at
worst subversive in relation to public concerns" (M. Brown, p. 263).
Without consensus on goals or criteria, without compelling results or
powerful theories, none of the rhetoric in these volumes can hope to
resolve researchers' continuing search for identity.

Thirty years ago the wonderland of category theory was in its heyday,
promising a "theory of theories" to unify mathematics. Floating in a
cloud far above equations, groups, and topologiesthe substance and
texture of mathematicscategory theorists promised powerful tools
and unifying theories. Traditionalists, skeptical as always, lampooned
the field as "generalized abstract nonsense." Category theory enlisted
a lot of machinery in the service of abstractions that were either
nonsense or perhaps every sense. No one knew for sure, but many had
their suspicions.

Research in mathematics education as reflected in these volumes is
much like category theory: a profusion of concepts, relations, and
neologisms brought to bear on abstract notions of mathematics education
that have none of the texture of real students in real classrooms. Many
readers will feel just like Alice: "Somehow it seems to fill my head
with ideasbut I don't exactly know what they are!" Whether real
educational improvement can emerge from these inchoate theories is far
from clear. But many have their suspicions.